Multiparticle Flux Tube S-matrix Bootstrap

TL;DR

This work extends the S-matrix bootstrap to the multi-particle sector of flux-tube dynamics by introducing branon jets—collinear, massless flux-tube excitations—as effective degrees of freedom. It defines a triplet of finite-energy observables (X, Y, Z) and demonstrates how sum rules bound their values, then constructs a nested Branon Matrioska of allowed S-matrix spaces under unitarity, analyticity, crossing, and low-energy EFT constraints. Through primal and dual semidefinite programming formulations, the authors show that imposing NG universality and an upper bound on the leading Wilson coefficient gamma drastically narrows the allowed region, yielding precise finite-energy predictions. The results illuminate the non-perturbative structure of flux tubes in 3D (and hint at rich behavior in higher dimensions), providing a concrete program to confront lattice data and guide future extensions to more complex multiparticle scattering.

Abstract

We introduce the notion of branon jets, states of collinear flux tube excitations. We argue for the analyticity, crossing and unitarity of the multi-particle scattering of these jets and, through the S-matrix bootstrap, place bounds on a set of finite energy multi-particle sum rules. Such bounds define a matrioska of sorts with a smaller and smaller allowed regions as we impose more constraints. The Yang-Mills flux tube, as well as other interesting flux tube theories recently studied through lattice simulations, lie inside a tiny island hundreds of times smaller than the most general space of allowed two-dimensional theories.

Figures (14)

• Figure 1: Different sum rules for different $n$'s lead to dramatically different integrands which nonetheless all integrate to exactly the same thing. Here we see that a low $n$ sum rule (on the left) decays slowly at large energies and is thus very sensitive to scattering there while a large $n$ sum rule (on the right) strongly suppresses high energy but requires a huge resolution at low energies which can also be a challenge. In the middle we depict a nice choice of intermediate $n=4$ which nicely suppresses large energy while not demanding a crazy resolution at low energy. In these plots the black and red curves represent two different Boostrap runs with two different $n_\text{max}$'s (130 and 75 respectively); we can think of them as two experiments with different resolution.
• Figure 2: The Branon Matrioska: allowed space of $(X,Y,Z)$ for various two dimensional theories where jets make sense. The blue shape assumes only unitarity and analyticity. Inside it is a much smaller red surface where we also impose non-linearly realized Lorentz by fixing the low energy behavior of the S-matrices. Inside it, in a yet much smaller green region is the space of such S-matrices with first Wilson coefficient $\gamma \lesssim 0.8$, a conservative upper bound which should contain most of the interesting flux tube theories according to the recent lattice estimates. In this figure we use $S_{n\to m}$ to denote the processes involving in total $n$ and $m$ particles in the initial and final states; $S_{4\to 4}$ for instance refers to the scattering of two jets (each with two particles) in the past yielding two jets in the future, that is $S_{4\to 4}=S_{22\to 22}=Y$ and so on. In each matrioska doll, some directions can be bounded from analytic single-component Schwarz type arguments, see appendix \ref{['analyticalbounds']}.
• Figure 3: At very low energy, the diagonal processes $11\to 11$ and $12\to 12$ dominate since the theory is free in the IR. As we crank up the energy jet production kicks in. On the even sector depicted on the left we see that around $s=4$ the jet production $11\to 22$ even starts dominating for this rightmost point of the red Matrioska. The solid curve on that left panel at $P=1$ is simply the sum over the two possible even outcomes. On the right we have the odd sector and that same sum no longer adds up to $1$! That means that around $s=1$ the optimal S-matrices at this point of the Matrioska choose to produce some odd state outside the $12$ branon/jet system! Would be fascinating to find out what these states are in case they have a physical meaning. This is the first instance of an S-matrix bootstrap where unitarity is perfectly saturated in a finite energy interval after which it is not. In all other examples we know of unitarity wants to converge to $1$. Note that in these plots we depicted two very different values of (very large) $n_\text{max}$ to be sure that there is no convergence issue.
• Figure S4: Particles of the same chirality do not scatter amongst each other. Indeed, we can always boost such configuration so that all particles would have very low energy and thus be effectively free.
• Figure S5: Leading Landau singularities come from either diagrams in which momentum cluster into subsets that scatter independently, such as the diagram in figure (1), or normal thresholds, such as the diagram in figure (2). In figure (2), $q_3$ is a left-mover, while $q_1$ and $q_2$ (and the remaining internal lines) are right movers. The loop equations imply $\alpha_3 q_3 =0$.